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Scattering Amplitudes in Quantum Field Theory

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Scattering Amplitudes in Quantum Field Theory Scattering Amplitudes in Quantum Field Theory European Physical Society Meeting Stockholm July 24, 2013 Zvi Bern, UCLA W q e º q g0 g g g g 1 Outline 1) Remarkable progress in scattering amplitudes. 2) Brief summary of new advances and ideas. 3) Example: applications to LHC physics. 4) Example: A duality between color and kinematics. 5) Example: UV surprises in supergravity theories. 2 Scattering amplitudes Scattering of elementary particles is fundamental to our ability to unravel microscopic laws of nature. Arrival of the Large Hadron Collider raises importance of collider physics and scattering amplitudes. Here we give some examples of advances of past few years in understanding and calculating scattering in quantum field theory. 3 Major Advance in Scattering Amplitudes “Impossible calculations” of scattering amplitudes in gauge and gravity theories now commonplace. A few highlights from past year: • Constructing large chunks of the scattering amplitudes of N = 4 super-Yang-Mills theory, towards a full construction. Alday, Arkani-Hamed, Basso, Bourjaily, Cachazo, Caron-Huot, Dixon, Duhr, Gehrmann, Golden, Goncharov, He, Henn, Heslop, Huber, Johansson, Kosower, Larsen, Lipstein, Lipatov, Maldacena, Mason, Pennington, Postnikov, Sikorowski, Sever, Spradlin, Trnka, Vergu, Vieira, Volovich and many others • New remarkable representations of gravity amplitudes inspired by twistor string theory. Adamo, Cheung, Hodges, Cachazo, Geyer, Mason, Skinner, etc • Advances in constructing string theory scattering amplitudes with large numbers of external legs. Broedel, Drummond, Green, Mafra, Schlotterer, Stieberger, Taylor, Ragousy, Terasoma, etc • Relations between gravity and gauge theory amplitudes. ZB, Bjerrum-Bohr, Carrasco, Davies, Dennen, O’Connell, Huang, Johansson ,Monteiro, Roiban , etc • NLO QCD multijet processes for LHC physics. See talk from de Florian ZB, Badger, Dixon, Febres Cordero, Hoeche, Ita,Kosower, Maitre, Ozeren, Uwer, Yundin, etc 4 Constructing Multiloop Amplitudes We do have powerful tools for computing amplitude. The ideas include: on-shell • Unitarity Method. ZB, Dixon, Dunbar, Kosower ZB, Carrasco, Johansson , Kosower • On-shell recursion. Britto, Cachazo Feng and Witten; Arkani-Hamed et al • Duality between color and kinematics. ZB, Carrasco and Johansson • Advances in loop integration technology. Chetyrkin, Kataev , Tkachov; Vladimirov; Marcus, Sagnotti; ZB, Dixon, Kosower; V.A. Smirnov; Czakon; Gehrmann, Remifdi; A.V. Smirnov; Britto, Cachazo, Feng; Bredenstein, Dennen, Dittmaier, Pozzorini; Ossola, Papadopoulos, Pittau; Forde; Badger ; ZB, Dixon, Kosower, Forde, Ita, Maitre; Ellis, Kunszt , Giele and many others. In this talk we will show you some applications of these ideas. 5 Example: Applications to NLO QCD and LHC Physics 6 G. Salam, ICHEP 2010 See Daniel de Florian’s talk 7 8 G. Salam, La Thuile 2012 2013: NLO W+5j [BlackHat+Sherpa: Bern et al] [unitarity] 9 2013: NLO 5j [Badger et al, Preliminary] [unitarity] ATLAS Comparison against NLO QCD ATLAS compared data against NLO theoretical predictions Powerful experimental confirmation of NLO approach. No tuning! Validate in regions where no new physics is expected and look for discrepancies on tails of distributions where new physics may be hiding. number of jets NLO predictions from BlackHat used to aid CMS search for supersymmetry W +1, 2, 3, 4 jets inclusive by giving reliable theory uncertainties. 10 W + 5 Jets in NLO QCD transverse hadronic energy spectrum A triumph for on-shell methods. Recently finished and accepted arXiv:1304.1253 in PRD LO/NLO ratio LO/NLO W q e º Transverse hadronic energy q g0 A new level for “state-of-the-art” LHC theory:g First NLO QCD 2 6 process. g g Seeg Daniel de Florian’s talk Example: A new understanding of gravity scattering amplitudes 12 Gravity vs Gauge Theory Consider the gravity Lagrangian flat metric graviton field metric Infinite number of complicated interactions + … terrible mess Compare to Yang-Mills Lagrangian on which QCD is based Only three and four point interactions Gravity seems so much more complicated than gauge theory. 13 Three Vertices Standard Feynman diagram approach. Three-gluon vertex: Three-graviton vertex: About 100 terms in three vertex Naïve conclusion: Gravity is a nasty mess. Definitely not a good approach. 14 Simplicity of Gravity Amplitudes People were looking at gravity the wrong way. On-shell formalism much more powerful. On-shell three vertices contains all information: gauge theory: double copy gravity: of Yang-Mills vertex. • Using modern on-shell methods, any gravity scattering amplitude constructible solely from on-shell 3 vertex. • Higher-point vertices irrelevant! BCFW recursion for trees, BDDK unitarity method for loops. 15 Duality Between Color and Kinematics momentum dependent coupling color factor kinematic factor constant Color factors based on a Lie algebra: Jacobi Identity Use 1 = s/s = t/t = u/u to assign 4-point diagram to others. n c n c n c tree = g2 s s + t t + u u A4 s t u Color factors³ satisfy Jacobi identity:´ Numerator factors satisfy similar identity: Color and kinematics satisfy the same identity 16 Duality Between Color and Kinematics Consider five-point tree amplitude: ZB, Carrasco, Johansson (BCJ) color factor kinematic numerator factor Feynman propagators a3a4b ba5c ca1a2 a3a4b ba2c ca1a5 a3a4b ba1c ca2a5 c1 f f f ; c2 f f f ; c3 f f f ´ ´ ´ c1 c2 + c3 = 0 n1 n2 + n3 = 0 ¡ ¡ Claim: At n-points we can always find a rearrangement so color and kinematics satisfy the same algebraic constraint equations. Nontrivial constraints on amplitudes in field theory and string theory BCJ, Bjerrum-Bohr, Feng,Damgaard, Vanhove, ; Mafra, Stieberger, Schlotterer; Cachazo; Tye and Zhang; Feng, Huang, Jia; Chen, Du, Feng; Du, Feng, Fu; Naculich, Nastase, Schnitzer 17 BCJ Gravity and Gauge Theory kinematic numerator color factor gauge sum over diagrams theory: with only 3 vertices Assume we have: c1 + c2 + c3 = 0 n1 + n2 + n3 = 0 , Then: ci n~i kinematic numerator of second gauge theory ) gravity: Gravity numerators are a double copy of gauge-theory ones! This works for ordinary Einstein gravity and susy versions! Cries out for a unified description of the sort given by string theory! Tree level proof, ZB, Dennen, Huang, Kiermaier; Bjerrum Bohr, Damgaard, Vanhove 18 Gravity loop integrands are free! BCJ Ideas generalize to loops: color factor kinematic (k) (i) (j) numerator If you can find a set of duality satisfying numerators. To get: gauge theory gravity theory simply take color factor kinematic numerator ck nk 19 Application: UV Properties of Gravity 20 Quantum Gravity Often repeated statement: “Einstein’s theory of General Relativity is incompatible with quantum mechanics.” To a large extent this is based on another often repeated statement: “All point-like quantum theories of gravity are ultraviolet divergent and non-renormalizable.” Where do these statements come from and are they true? 21 Power Counting at High Loop Orders Dimensionful coupling Gravity: Gauge theory: Extra powers of loop momenta in numerator means integrals are badly behaved in the UV. Non-renormalizable by power counting. Reasons to focus on N = 8 supegravity: • With more susy expect better UV properties. • High symmetry implies technical simplicity. 22 Finiteness of N = 8 Supergravity? If N = 8 supergravity is finite it would imply a new symmetry or non-trivial dynamical mechanism. No known symmetry can render a D = 4 gravity theory finite. The discovery of such a mechanism would have a fundamental impact on our understanding of gravity. Note: Perturbative finiteness is not the only issue for consistent gravity: Nonperturbative completions? High energy behavior of theory? Realistic models? Consensus opinion for the late 1970’s and early 1980’s: All supergravities would diverge by three loops and therefore not viable as fundamental theories. 23 Feynman Diagrams for Gravity SUPPOSE WE WANT TO CHECK IF CONSENSUS OPINION IS TRUE 20 3 loops ~10 No surprise it has never TERMS been calculated via − Calculations to settle Feynman diagrams. this seemed utterly hopeless! − Seemed destined for 4 loops ~1026 dustbin of undecidable TERMS questions. 31 5 loops ~10 More terms than TERMS atoms in your brain! 3- and 4-Loop Amplitude Construction ZB, Carrasco, Dixon, Johansson, Roiban In 2007 and 2010 using unitarity method. 3, 4 loop amplitudes constructed in N = 8 supergravity 3 loops: UV finite for D < 6 4 loops: UV finite for D < 11/2 These are very finite in D = 4 4 loops originally took more than a year. Today with the double copy we can reproduce it in a few days! 25 Current Status More recent papers argue that trouble starts at 5 loops and by 7 loops we have valid UV divergence in D = 4 under known symmetries. Bossard, Howe, Stelle; Elvang, Freedman, Kiermaier; Green, Russo, Vanhove ; Green and Bjornsson ; Bossard , Hillmann and Nicolai; Ramond and Kallosh; Broedel and Dixon; Elvang and Kiermaier; Beisert, Elvang, , Freedman, Kiermaier, Morales, Stieberger On the other hand duality between color and kinematics implies new constraints on the amplitudes. To settle the question it’s time to to calculate again. “Shut up and calculate!” 26 N = 8 Sugra 5 Loop Calculation ZB, Carrasco, Dixon, Johannson, Roiban A reasonable person would bet on divergences. But is a reasonable person right? Place your bets: 5 loops • At 5 loops in D = 24/5 does N = 8 supergravity diverge? •At 7 loops in D = 4 does N = 8 supergravity diverge? Kelly Stelle: Zvi Bern: English wine California wine “It will diverge” “It won’t diverge” 27 N = 4 Supergravity in D = 4 N = 4 sugra at 3 loops ideal D = 4 case to study. Cremmer, Scherk, Ferrara (1978) Consensus has N = 4 supergravity has a valid UV divergences in D = 4 under all known symmetries. Similar to N = 8 supergravity at 7 loops. Bossard, Howe, Stelle; Bossard, Howe, Stelle, Vanhove Is the consensus opinion true? BCJ representation exists for N = 4 sYM 3-loop 4-pt Amplitude. ZB, Carrasco, Johansson (2010) 28 The N = 4 Supergravity UV Cancellation ZB, Davies, Dennen, Huang (2012) N = 4 sugra = (N = 4 super-Yang-Mills)x(ordinary Yang-Mills) D = 4 2² ¡ All divergences cancel completely! UV finite contrary to expectations based on standard symmetries.
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